After all, the film has been a critical darling since it debuted at the Sundance Film Festival in January. Hammer and co. have been nominated for dozens of awards, and hope to win at the 2018Critics' Choice Awards and the 2018Independent Spirit Awards. Luca Guadagnino's movie, adapted from André Aciman's best-selling novel, is set in the Italian countryside in the early '80s. And the romantic coming-of-age drama, starring Golden Globe nominee Timothée Chalamet in the lead role, is expected to be a major player throughout the 2018 award season.

E! News spoke to Hammer by phone Monday, shortly after he learned of his nomination. "It's been good," he laughed. "It's always a good thing when your day peaks at 5:45 in the morning."

As it turned out, Hammer was already awake, as he has two young children at home. "I have to talk quietly, because Harper is still sleeping," the actor said apologetically. "But I can tell you how the conversation is going to go. She's going to come downstairs, and I'm going to say, 'Harper! Daddy got nominated for a Golden Globe!' She'll go, 'Oh. OK…Can I have pancakes for breakfast?'" Though his daughter may not be old enough to comprehend what a milestone this is, the 31-year-old actor sure is. "I've been in this business long enough that I know what a big deal it is, and what a great honor it is to get nominated for a Golden Globe. It's an amazing honor. It's great to have your work appreciated and recognized, and I wouldn't be here without Luca and Timmy and [co-star] Michael Stuhlbarg and [producer] Peter Spears, and André Aciman and [screenwriter] James Ivory. It's great, especially when you do something that you're proud of."

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Hammer hadn't had a chance to speak with Chalamet yet. "I tried calling him," he revealed, "but I think he's equally as busy as I am, doing phone calls. So, I haven't gotten ahold of him."

The stars have been promoting the film nearly a year, and now that it's receiving so much attention, expect to see a lot more of its stars. "It still feels like we're right in the middle of the promotion hurricane, especially when you get a Golden Globe nomination," Hammer said. "To get one for the film, and for Timmy as well, it feels like it's now going to kick us into high gear."

It hasn't been that long since principal photography began, even though it may feel like it. Hammer was initially hesitant to star in the film, but he decided to take a chance and put his faith in Guadagnino. "I had that moment [of clarity] when I got to Italy, really. The reality of what we were going to do and the undertaking that we had ahead of us kind of set in," he said. "That was really kind of a quick, passing feeling for me—that artist's insecurity kind of thing. But as soon as we got to Italy, I was like, 'I am so glad I did this. This was obviously the right move.'"

Now that so much time has passed since filming, he said, "I wish I was back in Italy. Oh, man! I've been back to visit just because I missed it so much. I went back to Crema—for no purpose!"

The cast and crew were fairly isolated when they shot Call Me by Your Name, which was foreign to Hammer, who had previously starred in big-budget, would-be blockbusters like The Lone Ranger and The Man From U.N.C.L.E. Going forward, he hopes to replicate the experience he had working on the indie. "There are definitely aspects that I would love to seek out—mostly just being challenged and working on art for the sake of art. But at the same time, every film is so different," he said. "I'm excited for how the next one is going to challenge me or push me, and how I'm going to react, deal with or engage with the next director that I work with and the next group of co-stars. It's always a fun process to get on a new project and kind of feel it out."

But before he lines up another role, his sights are on the Golden Globes (airing on NBC Jan. 7, 2018). Hammer is nominated in the same category as Willem Dafoe (The Florida Project), RichardJenkins (The Shape of Water), Christopher Plummer (All the Money in the World) and SamRockwell (Three Billboards Outside Ebbing, Missouri), and he is particularly fond of Jenkins' and Rockwell's performances. "I would be more than happy to lose to either of them. And I'm excited to see The Florida Project," Hammer said. "I haven't seen it yet but I heard it's amazing."

Even if Hammer doesn't walk away with a Golden Globe, he's just happy to be part of the film. "I already won by getting to be a part of this film. I already won because I got to work with Luca, I got to work with Timmy, I got to shoot a movie in the beautiful Italian countryside," he told E! News exclusively. "Anything that comes afterward would be icing on an already delicious cake."

Best of all, award season will provide plenty of opportunities for Hammer to mix and mingle with his favorite actors. "It's a wonderful time for actors to get together with their peers. I'm honestly really kind of excited about geeking out and being like, 'Man, Sam Rockwell! That was an amazing performance! Holy s--t, dude!' It's going to be fun to get to see all the people that you respect and admire their work," he explained. "It's always fun. Every time the Golden Globes comes around, you know that you're going to go to a great party. You know that you're going to have fun with a great collection of your peers. Bring it on! I'm sure I'm going to wake up in the morning feeling a little bit worse, because I am a little older now—but life goes on!"